polynomial computations in matlab · pdf fileabstract this report details how the following...

Polynomial Computations in MATLAB1

Kannan M. Moudgalya

Dept. of Chemical Engineering and

Systems and Control Group

Indian Institute of Technology

Powai, Mumbai 400 076

India

November 1996

1Supported by the Department of Science and Technology through Grant No. III.5(11)/91-ET

Abstract

This report details how the following polynomial matrix computations have been im-

plemented in MATLAB: coprime factorization, solutions to Aryabhatta’s Identity and

unilateral and bilateral polynomial equations. A complete listing of the source code is

provided. Usage of the software is illustrated with several examples.

Contents

1 Introduction 1

2 Left coprime factorization 3

2.1 Algorithm (Chang and Pearson, 1982) . . . . . . . . . . . . . . . . . . . 3

2.2 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.3 Example: Coprime factorization . . . . . . . . . . . . . . . . . . . . . . . 5

2.3.1 Input program listing (data01.m) . . . . . . . . . . . . . . . . . . 5

2.3.2 Output from MATLAB . . . . . . . . . . . . . . . . . . . . . . . . 6

2.3.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.4 Example: Checking whether a given polynomial matrix is unimodular . . 7

2.4.1 Input program listing . . . . . . . . . . . . . . . . . . . . . . . . . 7


2.5 Example: Producing a row proper polynomial . . . . . . . . . . . . . . . 8

2.5.1 Input program listing . . . . . . . . . . . . . . . . . . . . . . . . . 8


2.6 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3 Aryabhatta’s Identity 10


3.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.2.1 Input program listing (data04.m) . . . . . . . . . . . . . . . . . . 10

3.2.2 Output from MATLAB (ex04.out) . . . . . . . . . . . . . . . . . 11

3.2.3 Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.3 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4 Unilateral polynomial matrix equation 12


4.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.1 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2.2 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

4.3 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5 Generalized Aryabhatta’s Identity 16


5.1.1 Step1: Find A(s) and B(s) . . . . . . . . . . . . . . . . . . . . . . 16

5.1.2 Step2: Find A1(s), B1(s), X(s) and Y (s) . . . . . . . . . . . . . . 16

5.1.3 Step 3: Find X1(s) and Y1(s) . . . . . . . . . . . . . . . . . . . . 17

5.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

i

5.2.1 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

5.2.2 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

5.3 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

6 Bilateral polynomial matrix equation 22


6.2 Usage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2.1 First example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

6.2.2 Second example . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.3 Programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

7 Numerical issues 26

7.1 Linear dependence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

7.2 Zeroing small entries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

A MATLAB routines 28

A.1 arya.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.1.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.1.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.1.3 Listing of the code . . . . . . . . . . . . . . . . . . . . . . . . . . 28

A.2 cindep.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.2.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

A.2.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29


A.3 clcoef.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.3.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

A.3.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30


A.4 flat2nor.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.4.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

A.4.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31


A.5 indep.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

A.5.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

A.5.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32


A.6 issoln.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.6.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

A.6.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ii


A.7 l2r.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.7.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.7.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


A.8 left prm.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.8.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

A.8.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34


A.9 makezero.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A.9.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

A.9.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37


A.10 move.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.10.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.10.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


A.11 nor2flat.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.11.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

A.11.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


A.12 polksca.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.12.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

A.12.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39


A.13 scakpol.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.13.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.13.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


A.14 seshft.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.14.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

A.14.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40


A.15 t1calc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.15.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

A.15.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41


A.16 xdnyc.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

iii

A.16.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

A.16.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42


A.17 xdync.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.17.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

A.17.2 mfiles called . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43


iv

1 Introduction

In this report we present MATLAB version of programs to carry out polynomial cal-

cuations. The algorithms used here were proposed by Chang and Pearson (1982) and

implemented in FORTRAN by Chang (1982). Kwakernaak’s (1990) convention of repre-

senting polynomials in MATLAB has been adopted in this work.

In the control literature, the use of polynomial calculations has been reported in

discrete control in general (Kucera, 1979; Hunt, 1993; Grimble and Kucera, 1996; Astrom

and Wittenmark 1990), regulator theory (Cheng and Pearson, 1981), `1 optimal control

theory (Pearson, 1991), H∞ optimal control theory (Kwakernaak 1990) and behavioral

theory (Willems, 1991) in particular, to cite a few references. These theories require

coprime factorizations so often that a numerically stable implementation is indispensable.

The numerical stability of the current implementation is decided by the following two

calculations:

1. When does a set of vectors become dependent

2. What is a small number so that one can zero it

It is possible to improve the speed and the stability of the current implementation by

making the above two calculations more efficient. This report is organized as follows:

Section 2 describes the following computations:

• Given a transfer function matrix

H(s) = N(s)D(s)−1 (1)

N(s), D(s) not necessarily coprime, D(s) nonsingular, compute coprime polynomial

matrices A(s) and B(s) such that

A(s)−1B(s) = H(s) (2)

and A(s) row proper.

• Given two matrices N(s) and D(s) with D(s) nonsingular, check if N(s)D−1(s) is

a polynomial matrix.

• Given a polynomial matrix D(s), find a unimodular matrix W (s) and a row proper

D(s) so that

W (s)D(s) = D(s) (3)

1

Section 3 describes the following computation: Given coprime polynomial matrices

N(s) and D(s) with D(s) column proper, find polynomial matrices X(s) and Y (s) sat-

isfying the following identity.

X(s)D(s) + Y (s)N(s) = I (4)

This equation has been known as Bezout Identity (Kailath, 1980) and Aryabhatta’s Iden-

tity (Vidyasagar, 1985). In this report, we will refer to it as the Aryabhatta’s Identity.

Section 4 describes the following computation: Given polynomial matrices N(s), D(s)

and C(s) with D(s) nonsingular, find polynomial matrices X(s) and Y (s) satisfying

X(s)D(s) + Y (s)N(s) = C(s) (5)

if the above equation can be solved. This equation has been known as the Diophantine

Equation and Aryabhatta’s Identify in the literature. We will refer to it as the unilateral

polynomial matrix equation (Kucera, 1979).

Section 5 describes the following computation: Given arbitrary matrices N(s), D(s)

as given in Eq. 1, D(s) nonsingular, find the matrices in the following generalized Arya-

bhatta’s Identity

A(s) B(s)

−Y1(s) X1(s)

X(s) −B1(s)

Y (s) A1(s)

=

I 0

0 I

(6)

with

A(s)−1B(s) = B1(s)A1(s)−1 = H(s) (7)

Section 6 describes the following computation: Given polynomial matrices N(s), D(s)

and C(s) with D(s) nonsingular, find polynomial matrices X(s) and Y (s) satisfying

X(s)D(s) + N(s)Y (s) = C(s) (8)

if the above equation can be solved. This equation is known as the bilateral polynomial

matrix equation (Kucera, 1979).

Section 7 is devoted to numerical issues, such as, when a set of vectors is dependent

and how to decide whether a given number is small.

2

2 Left coprime factorization

In this section we will be concerned with the calculation of B(s) and A(s) matrices

satisfying Eq. 2, given matrices satisfying Eq. 1. Equivalently, we need to compute

[−B(s) A(s)]

D(s)

N(s)

= 0 (9)

2.1 Algorithm (Chang and Pearson, 1982)

Define

F (s) =

D(s)

N(s)

(10)

Let

F (s) = F0 + F1s + F2s2 + · · · + Fwsw (11)

where, Fi is the matrix of coefficients of si in the polynomial matrix F(s). It is assumed

that the degree of F (s) is w. We want to compute the Modified Minimal Basis E1(s),

where

E1(s) = [−B(s) A(s)] (12)

such that

E1(s)F (s) = 0 (13)

Assume that E1(s) is a polynomial matrix of degree v, which is to be determined:

E1(s) = E10 + E11s + E12s2 + · · · + E1vs

v (14)

Equating the coefficients of powers of s in Eq. 13, we get

T1 S = 0 (15)

where

T1 = [E10 E11 E12 · · · E1v] (16)

and

S =

F0 F1 · · · Fw 0 0 · · · 0 0

0 F0 F1 · · · Fw 0 · · · 0 0...

0 0 0 · · · 0 F0 F1 · · · Fw

(17)

Since D(s) is nonsingular, the first row of S is nonzero.

Let the kth row of F be denoted by f ′k. Find the maximum k such that the first k

rows are independent. In this case, let

b11f′1 + b12f

′2 + · · · + b1k1f

′k1

+ f ′k1+1 = 0 (18)

3

Then the first row of T1 is

[b11 b12 · · · b1k1 1 0 · · · 0]

Since the (k1 + 1)st entry is fixed at 1, the above row is unique.

We next proceed to compute the second row of T1. First of all, one of the first k1 +1

rows of S has to be deleted to make the remaining k1 rows linearly independent. The

row to be deleted is decided as follows: From Eq. 12 and Eq. 16,

T1 = [−B0 A0 −B1 A1 · · · −Bv Av] (19)

Define a set of integers in which every integer corresponds to a column of A0, A1, · · ·in T1. This set is denoted by a and

a = {i : integer, n ≤ (i− 1)(mod q) ≤ n + m− 1} (20)

where n and m come from the dimensions of N(s) and D(s) and q = m + n:

N(s) ∈ Rm×n[s] and D(s) ∈ Rn×n[s].

Choose the largest integer ≤ k1 + 1 and belonging to a. Call this pr(1), primary

redundant row. Delete the following rows of S: pr(1), pr(1) + q, pr(1) + 2q, · · ·. that is,

delete every qth row starting with pr(1)th. The rows that are deleted will be known as

redundant rows.

Begin with k = k1 + 1 and test if f ′k+1 is a linear combination of previous

independent rows in S, where S denotes the S matrix with redundant rows delted.

Increase k and continue this test until an integer k2 is found such that f ′k2+1 is a linear

combination of the previous rows in S. Then the second row in T1 can be determined.

In this row, the coefficients that multiply the redundant rows and rows > k2 + 1 of S

matrix will be zero.

The next primary redundant row pr(2) is one of the dependent rows used in the

calculation of the second row of T1. pr(2) is the largest number ≤ k2 + 1 and belonging

to a. If pr(2) < pr(1), replace pr(1) with pr(2) and recalculate the second row of T1.

After pr(2) is determined, all redundant rows of S matrix are deleted.

This procedure is repeated until all required rows of T1 are determined. Because of

the assumed dimensions for N(s) and D(s), there should be m rows in T1. From Eq. 12,

Eq. 14 and Eq. 16, it is clear that by splitting the T1 matrix appropriately, we can get

−B(s) and A(s).

2.2 Conventions

Kwakernaak’s (1991) convention of writing the coefficient matricies corresponding to

ascending powers of s is used in this work. For example,

N(s) =

0 4 + s

−1 8 + 3s

=

0 4

−1 8

+

0 1

0 3

s (21)

4

will have to be entered in MATLAB as,

N =

0 4 0 1

−1 8 0 3

(22)

The second convention is that every polynomial matrix will have two variables: one will

have the coefficient matrix and the other the degree of the polynomial matrix. In the

above example, for instance, the degree of N(s) is one. Unless this is specified, the

number of columns of N(s) cannot be determined. The third convention is that the

coefficient matrix will be entered first and then its degree.

2.3 Example: Coprime factorization

In this section we discuss the usage of the program with an example. Let

N(s) =

1 0 0

0 1 0

0 0 1

and D(s) =

1 0 0

s 1 0

s s 1

(23)

2.3.1 Input program listing (data01.m)

An m file containing the following statements is executed.

format compact

format short

D = [

1 0 0 0 0 0

0 1 0 1 0 0

0 0 1 1 1 0]

N = [

1 0 0

0 1 0

0 0 1]

degD = 1

degN = 0

[Lnum,degLnum,Lden,degLden] = left_prm(N,degN,D,degD)

Notice that there are only four input arguments to the function-call left prm. The symbol

Lnum stands for left numerator and similarly Lden stands for left denominator. The prefix

deg stands for degree.

5

2.3.2 Output from MATLAB

The following is the resulting output from MATLAB (ex01.out).

D =

1 0 0 0 0 0

0 1 0 1 0 0

0 0 1 1 1 0

N =

1 0 0

0 1 0

0 0 1

degD =

1

degN =

0

Lnum =

Columns 1 through 7

1.0000 0 0 0 0 0 0

0 1.0000 0 -1.0000 0 0 0

0 0 -1.0000 1.0000 1.0000 0 -1.0000

Columns 8 through 9

0 0

0 0

0 0

degLnum =

2

Lden =

1 0 0

0 1 0

0 0 -1

degLden =

0

2.3.3 Result

For N(s) and D(s) as given in Eq. 23, B(s) and A(s) satisfying Eq. 9 are as given below:

B(s) =

1 0 0

−s 1 0

s− s2 s −1

and A(s) =

1 0 0

0 1 0

0 0 −1

(24)

6

2.4 Example: Checking whether a given polynomial matrix is

unimodular

Suppose we want to decide whether a given nonsingular matrix D(s) is unimodular.

Carry out left coprime factorization as given below:

A(s)−1B(s) = I D(s)−1 (25)

Where I is an identity matrix and A is row proper. It is clear that D is unimodular if

and only if A is a constant matrix. This is illustrated with a unimodular D matrix.

2.4.1 Input program listing

The following MATLAB mfile is executed (data02.m).

format compact

format short

D = [1 0 0 0 0 0

0 1 0 1 0 0

0 0 1 1 1 0]

degD = 1

[B,degB,A,degA] = left_prm(eye(3),0,D,degD)

if degA == 0,

’Given matrix D is unimodular’

end


The following is the resulting output (ex02.out).

D =

1 0 0 0 0 0

0 1 0 1 0 0

0 0 1 1 1 0

degD =

1

B =

Columns 1 through 7

1.0000 0 0 0 0 0 0

0 1.0000 0 -1.0000 0 0 0

0 0 -1.0000 1.0000 1.0000 0 -1.0000

Columns 8 through 9

7

0 0

0 0

0 0

degB =

2

A =

1 0 0

0 1 0

0 0 -1

degA =

0

ans =

Given matrix D is unimodular

2.5 Example: Producing a row proper polynomial

Given a nonsingular polynomial matrix D(s), we wish to find a unimodular matrix W (s),

such that

W (s)D(s) = D(s) (26)

where D(s) is row proper. This is equivalent to finding a left coprime pair D(s), W (s),

such that

D(s)−1W (s) = I D(s)−1 (27)

2.5.1 Input program listing

The following MATLAB mfile is executed (data03.m).

format compact; format short

D = [4 0 12 0 13 0 6 0 1 0

2 2 5 1 4 0 1 0 0 0]

degD = 4

[W,degW,Dhat,degDhat] = left_prm(eye(2),0,D,degD)


The following is the resulting output (ex03.out).

D =

4 0 12 0 13 0 6 0 1 0

2 2 5 1 4 0 1 0 0 0

degD =

8

4

W =

-1.0000 2.0000 0 1.0000

0 1.0000 0 0

degW =

1

Dhat =

Columns 1 through 7

0 4.0000 0 4.0000 0 1.0000 0

2.0000 2.0000 5.0000 1.0000 4.0000 0 1.0000

Column 8

0

0

degDhat =

3

2.6 Programs

The following programs, whose listings appear in alphabetical order in the appendix, are

used in the above-said calculation:

1. left prm.m

2. indep.m

3. makezero.m

4. move.m

5. seshft.m

6. t1calc.m

The user starts the calculation by invoking left prm.m. The following routines of Kwak-

ernaak (1990) are also required:

1. clcoef.m

2. colsplit.m

3. polsize.m

4. rowjoin.m

The program clcoef.m has been modified in this work and its listing also appears in the

appendix. For other routines, the reader is referred to Kwakernaak (1990).

9

3 Aryabhatta’s Identity

In this section, we will find polynomial matrices X(s) and Y (s) satisfying the Arya-

bhatta’s Identity,

X(s)D(s) + Y (s)N(s) = I (28)

given coprime polynomial matrices N(s) and D(s) with D(s) column proper.


Given N(s) and D(s) as given above, go through the entire calculation of Sec. 2 to

compute T1 matrix of Eq. 15 and Eq. 16. In this process, left coprime matrices B(s) and

A(s) can also be computed without much more effort.

As it is given that N(s) and D(s) are coprime with D(s) column proper, there exist

X(s) and Y (s) satisfying Eq. 28 with deg[X(s) Y (s)] < deg[−B(s) A(s)].

Let S denote S matrix of Eq. 17 with all redundant rows deleted. Thus S contains

only independent rows. Thus it is possible to solve

b S = I (29)

After b is found, it is augmented with zero columns at locations corresponding to the

rows of S deleted. Call the matrix thus obtained as T2. As the polynomial equivalent of

this matrix satisfies

T2(s) = [X(s) Y (s)] (30)

X(s) and Y (s) can be calculated.

3.2 Usage

We now illustrate how to use this program with an example. Let

N(s) =

0 4 + s

−1 8 + 3s

and D(s) =

−s 4s + s2

s 0

(31)

3.2.1 Input program listing (data04.m)

An m file containing the following statements is executed.

format compact

format short

D = [

0 0 -1 4 0 1

0 0 1 0 0 0]

10

N = [

0 4 0 1

-1 8 0 3 ]

degD = 2

degN = 1

[Lnum,degLnum,Lden,degLden,Y,degY,X,degX] = left_prm(N,degN,D,degD,2)

Notice that we use the routine left prm.m once again as we did in Sec. 2.3.1. The

same naming convention of variables is used. Notice, however, that there are five input

arguments used in the call to left prm while only four are required in Sec. 2.3.1. The

fifth argument to left prm could be one, two or three. If it is one, coprime factorization

is done. If it is two, Aryabhatta’s Identity is solved. Finally, if it is three, it will try to

solve Eq. 5. In this case, C(s) also required as an input. This is the topic of discussion

in Sec. 4.

3.2.2 Output from MATLAB (ex04.out)

D =

0 0 -1 4 0 1

0 0 1 0 0 0

N =

0 4 0 1

-1 8 0 3

degD =

2

degN =

1

Lnum =

1.0000 1.0000 0 0

8.0000 4.0000 3.0000 2.0000

degLnum =

1

Lden =

0 0 1.0000 0 0 0

0 0 0 4.0000 0 1.0000

degLden =

2

Y =

2.0000 -1.0000 0 -0.2500

0.2500 0 0 0.0625

11

degY =

1

X =

0.7500 0.5000

-0.1875 -0.1250

degX =

0

3.2.3 Result

The results can be summarized as:

B(s) =

1 1

8 + 3s 4 + 2s

A(s) =

s 0

0 4s + s2

(32)

and

X(s) =

0.75 0.5

−0.1875 −0.125

Y (s) =

2 −1− 0.25s

0.25 0.0625

(33)

3.3 Programs

Same as in Sec. 2.6. The user calls left prm.m.

4 Unilateral polynomial matrix equation

In this section, we will find polynomial matrices X(s) and Y (s) satisfying

X(s)D(s) + Y (s)N(s) = C(s) (34)

given polynomial matrices N(s), D(s) and C(s) with D(s) column proper.


There is a solution to Eq. 34 if and only if the greatest common right divisor of D(s) and

N(s) is also a right divisor of C(s).

Let B1(s)A1(s)−1 be a right coprime factorization of N(s)D(s)−1, R(s)−1U1(s) a left

coprime factorization of D(s)−1A1(s) and U2(s)−1C2(s) a left coprime factorization of

C(s)R(s)−1. Then Eq. 34 has a solution if and only if U2(s) is row proper and constant.

Once it is determined that a solution exists to Eq. 34, one can solve it using the same

technique used for solving Eq. 28, as discussed in Sec. 3.1. That is, we only have to

substitue C(s) in the place of I.

12

4.2 Usage

We will illustrate the solution to Eq. 34 with two examples.

4.2.1 First example

An m file containing the following statements is executed (data05.m).

format compact

format short

Num = [0 4 0 1

-1 8 0 3]

degNum = 1

Den = [0 0 1 4 0 1

0 0 -1 0 0 0]

degDen = 2

C = [1 0 1 1

0 2 0 1]

degC = 1

[Lnum,degLnum,Lden,degLden,Y,degY,X,degX] = ...

xdync(Num,degNum,Den,degDen,C,degC)

and the following MATLAB results are obtained (ex05.out):

Num =

0 4 0 1

-1 8 0 3

degNum =

1

Den =

0 0 1 4 0 1

0 0 -1 0 0 0

degDen =

2

C =

1 0 1 1

0 2 0 1

degC =

1

Lnum =

1.0000 1.0000 0 0

13

8.0000 12.0000 3.0000 4.0000

degLnum =

1

Lden =

0 0 1 0 0 0

0 0 0 4 0 1

degLden =

2

Y =

2.0000 -1.0000 0 -0.5000

0.5000 0 0 -0.1250

degY =

1

X =

1.5000 1.0000

0.3750 0.5000

degX =

0

4.2.2 Second example


format compact

format short e

Num = [0 0.828 72.16 0.556

0 0.23 18.05 0.03 ]

degNum = 1

Den = [-1 0 1 0

0 -1 0 1]

degDen = 1

C = [4.002778 0.09749997

-3006.672 -95.99581]

degC = 0

[Lnum,degLnum,Lden,degLden,Y,degY,X,degX] = ...

xdync(Num,degNum,Den,degDen,C,degC)


Num =

0 8.2800e-001 7.2160e+001 5.5600e-001

14

0 2.3000e-001 1.8050e+001 3.0000e-002

degNum =

1

Den =

-1 0 1 0

0 -1 0 1

degDen =

1

C =

4.0028e+000 9.7500e-002

-3.0067e+003 -9.5996e+001

degC =

0

Lnum =

0 8.2800e-001 7.2160e+001 5.5600e-001

0 2.3000e-001 1.8050e+001 3.0000e-002

degLnum =

1

Lden =

-1.0000e+000 0 1.0000e+000 0

0 -1.0000e+000 0 1.0000e+000

degLden =

1

Y =

1.1563e-001 -2.4049e-001

-1.5290e+002 4.4469e+002

degY =

0

X =

-4.0028e+000 -5.7074e-002

3.0067e+003 7.1673e+001

degX =

0

4.3 Programs

In addition to the programs mentioned in Sec. 2.6, we also need

1. cindep.m

15

2. issoln.m

3. xdync.m

The user starts the calculations with xdync.m

5 Generalized Aryabhatta’s Identity

Given matrices as given in Eq. 35, N(s), D(s) not necessarily coprime, D(s) nonsingular,

H(s) = N(s)D(s)−1 (35)

find matrices in the following generalized Aryabhatta’s Identity: A(s) B(s)

−Y1(s) X1(s)

X(s) −B1(s)

Y (s) A1(s)

=

I 0

0 I

(36)

with

A(s)−1B(s) = B1(s)A1(s)−1 = H(s) (37)


Using the algorithms developed earlier, one can compute the required matrices in three

steps:

5.1.1 Step1: Find A(s) and B(s)

Given H(s) = N(s)D(s)−1 find A(s) and B(s) satisfying Eq. 37. This is done using the

procedure explained in Sec. 2. A(s) and B(s) are left coprime and A(s) is row proper.

5.1.2 Step2: Find A1(s), B1(s), X(s) and Y (s)

Starting from A(s) and B(s) left coprime, A(s) row proper, calculate right coprime

A1(s) and B1(s) satisfying Eq. 37 with A1(s) column proper. Also solve the following

Aryabhatta’s Identity for X(s) and Y (s):

A(s)X(s) + B(s)Y (s) = I (38)

These calculations are carried out by transposing A(s) and B(s), using the procedure

outlined in Sec. 2 and Sec. 3 and transposing the resulting matrices.

16

5.1.3 Step 3: Find X1(s) and Y1(s)

Using the procedure outlined in Sec. 3, find X1(s) and Y1(s) satisfying the following

Aryabhatta’s idenity:

X1(s)A1(s) + Y1(s)B1(s) = I (39)

5.2 Usage


5.2.1 First example


format compact

format short e

Num = [361 72.3 .2 -17.95 0 0

5.2 .18 2.88 .03 .46 0]

degNum = 2

Den = [10 0 6 0 1 0

0 10 0 6 0 1]

degDen = 2

[A,degA,B,degB,A1,degA1,B1,degB1,X,degX,Y,degY,X1,degX1,Y1,degY1] = ...

arya(Num,degNum,Den,degDen)


Num =

3.6100e+002 7.2300e+001 2.0000e-001 -1.7950e+001 0 0

5.2000e+000 1.8000e-001 2.8800e+000 3.0000e-002 4.6000e-001 0

degNum =

2

Den =

10 0 6 0 1 0

0 10 0 6 0 1

degDen =

2

A =

5.0028e+000 -3.0067e+003 1.0000e+000 0

1.6667e-003 9.9722e-001 0 1.0000e+000

degA =

17

1

B =

-1.3829e+003 -1.7950e+001 0 0

5.7872e-001 3.0000e-002 4.6000e-001 0

degB =

1

A1 =

-4.0000e+000 -2.5000e+000 1.0000e+000 0

2.0000e+001 1.0000e+001 0 1.0000e+000

degA1 =

1

B1 =

2.0000e-001 -1.7950e+001 0 0

-1.7200e+000 -1.1200e+000 4.6000e-001 0

degB1 =

1

X =

0 0

-6.3889e-003 -3.8227e+000

degX =

0

Y =

1.3889e-002 8.3102e+000

-5.5556e-002 9.2593e-002

degY =

0

X1 =

-3.8227e+000 0

-4.2593e-002 0

degX1 =

0

Y1 =

1.3889e-002 8.3102e+000

-5.5556e-002 9.2593e-002

degY1 =

0

18



format compact

format short e

Num = [

675 90 70 100 720 228 139 185 282 192 90 108 48 60 23 25 3 6 2 2

16 40 200 128 28 78 130 160 14 49 28 64 2 12 2 8 0 1 0 0

288 0 40 544 328 -80 78 840 136 -76 49 472 26 -22 12 114 2 -2 1 10]

Den = [

120 0 0 0 274 0 0 0 225 0 0 0 85 0 0 0 15 0 0 0 1 0 0 0

0 120 0 0 0 274 0 0 0 225 0 0 0 85 0 0 0 15 0 0 0 1 0 0

0 0 120 0 0 0 274 0 0 0 225 0 0 0 85 0 0 0 15 0 0 0 1 0

0 0 0 120 0 0 0 274 0 0 0 225 0 0 0 85 0 0 0 15 0 0 0 1]

degNum = 4

degDen = 5

[A,degA,B,degB,A1,degA1,B1,degB1,X,degX,Y,degY,X1,degX1,Y1,degY1] = ...

arya(Num,degNum,Den,degDen)


Num =

Columns 1 through 12

675 90 70 100 720 228 139 185 282 192 90 108

16 40 200 128 28 78 130 160 14 49 28 64

288 0 40 544 328 -80 78 840 136 -76 49 472


48 60 23 25 3 6 2 2

2 12 2 8 0 1 0 0

26 -22 12 114 2 -2 1 10

Den =


120 0 0 0 274 0 0 0 225 0 0 0

0 120 0 0 0 274 0 0 0 225 0 0

0 0 120 0 0 0 274 0 0 0 225 0

0 0 0 120 0 0 0 274 0 0 0 225


85 0 0 0 15 0 0 0 1 0 0 0

0 85 0 0 0 15 0 0 0 1 0 0

0 0 85 0 0 0 15 0 0 0 1 0

19

0 0 0 85 0 0 0 15 0 0 0 1

degNum =

4

degDen =

5

A =

Columns 1 through 6

8.0000e+000 -8.5714e-001 -6.0714e-001 1.4000e+001 -1.2857e+000 -8.5714e-001

0 6.0000e+000 -4.5000e+000 0 1.1000e+001 -6.0000e+000

0 0 1.5000e+001 0 0 2.3000e+001


7.0000e+000 -4.2857e-001 -2.5000e-001 1.0000e+000 0 0

0 6.0000e+000 -1.5000e+000 0 1.0000e+000 0

0 0 9.0000e+000 0 0 1.0000e+000

degA =

3

B =

Columns 1 through 6

4.3429e+001 5.7143e+000 3.0357e+000 3.0000e+000 2.3500e+001 1.2071e+001

-1.0000e+001 2.0000e+000 8.5000e+000 -1.4000e+001 -1.0000e+000 6.0000e+000

3.6000e+001 0 5.0000e+000 6.8000e+001 1.4000e+001 -1.0000e+001


6.7500e+000 6.5000e+000 3.0000e+000 6.0000e+000 2.0000e+000 2.0000e+000

5.0000e-001 -7.0000e+000 0 1.0000e+000 0 0

6.0000e+000 5.4000e+001 2.0000e+000 -2.0000e+000 1.0000e+000 1.0000e+001

degB =

2

A1 =

Columns 1 through 6

-2.3745e+000 3.6943e+000 -8.0572e+000 1.3912e+001 -2.8235e+000 4.8235e+000

3.5090e+000 3.8856e+000 -1.0644e+001 9.7473e+000 4.8547e+000 1.0908e+000

-5.7623e-002 2.4658e+000 -1.3421e+000 1.2173e+000 -2.8812e-002 3.2329e+000

1.8727e-001 -5.1381e-001 7.3619e+000 -3.9562e+000 7.2029e-002 4.1777e-001


-1.0265e+001 2.1147e+001 -4.4898e-001 1.1293e+000 -2.2075e+000 8.2347e+000

-2.5916e+000 2.4436e+000 1.0000e+000 0 0 0

-6.7107e-001 6.0864e-001 0 1.0000e+000 0 0

5.6777e+000 -1.5216e+000 0 0 1.0000e+000 0

20


0 0 0 1.0000e+000

0 0 0 0

0 0 0 0

0 0 0 0

degA1 =

3

B1 =

Columns 1 through 6

-1.0603e+001 2.4705e+001 -4.7953e+001 8.2981e+001 4.6531e+000 5.3878e+000

9.5678e-001 5.3493e+000 9.9340e-001 2.9130e+000 1.0000e+000 0

-4.8691e+000 7.3589e+000 1.3589e+001 1.5861e+001 -2.8980e+000 3.2585e+000


-4.6224e+000 2.7704e+001 0 0 0 3.0000e+000

0 2.0000e+000 0 0 0 0

5.5850e+000 1.2469e+001 0 0 0 2.0000e+000

degB1 =

2

X =

1.0422e-001 6.7621e-001 4.1309e-001 1.0364e-001 7.3289e-002 -8.5167e-003

1.4550e-002 6.0859e-002 2.5528e-002 0 0 0

7.3995e-001 1.6999e-001 -2.7790e-001 6.9090e-002 4.8859e-002 -5.6778e-003

degX =

1

Y =

Columns 1 through 6

-8.0407e-003 -1.1722e-001 -6.7598e-002 -8.4252e-002 -1.5058e-001 -6.9968e-002

2.2981e-001 -7.9226e-002 -1.3647e-001 5.4541e-002 -1.2000e-002 -3.1206e-002

5.0045e-002 7.6593e-002 -2.5357e-002 2.5023e-002 -4.5037e-002 -5.4345e-002

-1.6265e-001 1.8929e-002 1.1366e-001 -6.2556e-002 5.4491e-003 4.2113e-002

Columns 7 through 9

-3.4545e-002 -2.4430e-002 2.8389e-003

0 0 0

0 0 0

0 0 0

degY =

2

X1 =

21

Columns 1 through 6

-7.8991e-001 1.9245e-001 -7.1474e-002 -2.3854e-001 -1.0121e-001 -3.7766e-001

-7.3473e-002 -1.6233e-001 1.0418e-001 7.9751e-001 3.3623e-002 -2.1379e-001

7.0154e-001 2.2725e-001 1.1862e-002 -1.1188e+000 1.0344e-001 4.5411e-001

4.7862e-001 2.2771e-001 6.4675e-002 -6.8624e-001 9.7957e-002 2.3738e-001

Columns 7 through 8

-7.7875e-002 2.0298e-001

4.3002e-003 4.9341e-001

8.3000e-002 -2.9602e-001

6.6251e-002 4.0701e-002

degX1 =

1

Y1 =

Columns 1 through 6

8.3183e-002 -1.1490e-001 1.0650e-001 2.4181e-001 6.6016e-003 -9.9980e-002

1.0534e-001 2.2385e-001 -2.3711e-001 1.2492e-001 -4.0572e-002 -2.5184e-001

-2.6336e-001 -2.3916e-002 3.4724e-001 -3.1231e-001 3.0003e-002 2.9031e-001

-1.7092e-001 -1.4296e-001 2.4762e-001 -1.8346e-001 -2.3686e-002 1.1781e-001

Columns 7 through 9

5.4541e-002 -1.2000e-002 -3.1206e-002

2.5023e-002 -4.5037e-002 -5.4345e-002

-6.2556e-002 5.4491e-003 4.2113e-002

-3.4545e-002 -2.4430e-002 2.8389e-003

degY1 =

2

5.3 Programs

In addition to the programs mentioned in Sec. 2.6, we also need arya.m and transp.m

developed by Kwakernaak (1990). The calculations are started by executing arya.m.

6 Bilateral polynomial matrix equation

In this section, we will find polynomial matrices X(s) and Y (s) satisfying

X(s)D(s) + N(s)Y (s) = C(s) (40)

given polynomial matrices N(s), D(s) and C(s) with D(s) column proper and the fol-

lowing dimensions: N(s) ∈ Rm×h[s], D(s) ∈ Rn×n[s] and C(s) ∈ Rm×n[s].

22


Let x′k, y′k and c′k be the kth row vectors of X(s), Y (s) and C(s) respectively, i.e.

x′k = [xk1 xk2 . . . xkn] k = 1, . . . ,m (41)

y′k = [yk1 yk2 . . . ykn] k = 1, . . . , h (42)

c′k = [ck1 ck2 . . . ckn] k = 1, . . . ,m (43)

where xkj is the element of X(s) at k-th row and j-th column. ykj and ckj are defined in

a similar way. Then Eq. 40 can be transformed to

X(s)D(s) + Y (s)N(s) = C(s) (44)

with the following definitions:

X = [x′1 x′2 . . . x′m] (45)

Y = [y′1 y′2 . . . y′h] (46)

C = [c′1 c′2 . . . c′m] (47)

D = Im ⊗ D (48)

N = N ′ ⊗ In (49)

where ⊗ denotes Kronecker product.

Now Eq. 44 is of the form Eq. 34. Thus the algorithm used in Sec. 4 can be used for

this purpose.

6.2 Usage


6.2.1 First example


format compact

format short

Num = [-16 8 0 4

-4 0 0 1]

degNum = 1

Den = [0 0 -1 4 0 1

0 0 1 0 0 0]

degDen = 2

C = [1 0 1 1

23

0 2 0 1]

degC = 1

[Y,degY,X,degX] = xdnyc(Num,degNum,Den,degDen,C,degC)


Num =

-16 8 0 4

-4 0 0 1

degNum =

1

Den =

0 0 -1 4 0 1

0 0 1 0 0 0

degDen =

2

C =

1 0 1 1

0 2 0 1

degC =

1

Y =

0 -0.5000 0 -0.1250

0.1250 -1.0000 0 -0.3750

degY =

1

X =

1.5000 2.0000

0.3750 0.2500

degX =

0



format compact

format short

Num = [

1 0

1 1]

24

degNum = 0

Den = [

1 0 0 0 0 -1 0 0 0 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 -2 0 0 0 0 1 0

0 0 0 0 1 0 0 0 0 -1 0 0 0 0 0]

degDen = 2

C = [

0 0 0 -3 0 1 0 0 2 0

0 0 0 0 -1 0 0 0 0 0]

degC = 1

[Y,degY,X,degX] = xdnyc(Num,degNum,Den,degDen,C,degC)


Num =

1 0

1 1

degNum =

0

Den =


1 0 0 0 0 -1 0 0 0 0 0 0

0 1 0 0 0 0 0 0 0 0 0 0

0 0 1 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 -2 0 0 0

0 0 0 0 1 0 0 0 0 -1 0 0


0 0 0

0 0 0

0 0 0

0 1 0

0 0 0

degDen =

2

C =

0 0 0 -3 0 1 0 0 2 0

0 0 0 0 -1 0 0 0 0 0

degC =

25

1

Y =

Columns 1 through 7

1.0000 0 0 -3.0000 0 0 0

-1.0000 0 0 3.0000 -1.0000 0 0


0 2.0000 0

0 -2.0000 0

degY =

1

X =

-1.0000 0 0 0 0

0 0 0 0 0

degX =

0

6.3 Programs

In addition to the programs mentioned in Sec. 2.6, we also need the following:

1. flat2nor.m

2. nor2flat.m

3. polksca.m

4. scakpol.m

5. xdync.m

and transp.m developed by Kwakernaak (1990). The user initiates the calculations with

xdync.m.

7 Numerical issues

The most crucial step in computing coprime factorization is the determination of lin-

ear dependence. We have used the singular values to detect the occurrence of linear

dependence. We also use a zeroing strategy to make small terms zero.

26

7.1 Linear dependence

Suppose we want to check whether the mth row of an m× n matrix A is dependent on

the previous m−1 rows. It is assumed that the first m−1 rows are linearly independent.

Compute the singular values of A. Let σ be the maximum and σ be the minimum of

these. If

σ/σ < ε×max(m, n)

then mth row is dependent on the rest. Here ε is the machine epsilon. Suppose the above

condition is violated. Then we carry out the following test to confirm that the rows are

independent. Let the singular values be arranged in descending order. If the ratio of

the successive singular values is larger than a large number G, we take the rows to be

independent. In our calculations, we have taken the default value of G to be 108, but

the user can override it. This is implemented in the routines that check independence,

namely indep.m, Sec. A.5 and cindep.m, Sec. A.2.

7.2 Zeroing small entries

Small entries of a vector are made zero with the procedure as suggested by Chang, (1982).

The absolute values of the vector are arranged in descending order. If the ratio of two

successive entries is greater than a large number G, we zero all succeeding entries, starting

from the smaller of the two. We have chosen this large number to the same as the one

discussed in the previous sub-section. This is implemented in Sec. A.9.

27

A MATLAB routines

All the m files developed in this work are listed in this section. These are listed under the

heading local routines . We have listed two routines from the work of Kwakernaak (1990)

because we have modified them in the current work. These are clcoef.m and l2r.m. In

the following, we also list the names of functions called by each routine. We have also

listed the functions that came with MATLAB. These have been listed under the heading

Routines of mathworks.

A.1 arya.m

A.1.1 Purpose

Used to solve generalized Aryabhatta’s Identity, Eq. 36.

A.1.2 mfiles called

Local routines: left prm.

Kwakernaak’s routines: transp.

A.1.3 Listing of the code

% function ..

% [A,degA,B,degB,A1,degA1,B1,degB1,X,degX,Y,degY,X1,degX1,Y1,degY1] ...

% = arya(N,degN,D,degD)

% calculates the entries of generalized aryabhatta identity

% -1

% given H = N * D not necessarily coprime,

% finds A, B, A1, B1, X, Y, X1 and Y1 such that V * W = I, where

%

% ( A B ) ( X -B1 )

% V = ( ) and W = ( )

% ( -Y1 X1 ) ( Y A1 )

% -1 -1

% and H = A * B = B1 * A1 , A, B left coprime and B1, A1 right coprime

%

function [A,degA,B,degB,A1,degA1,B1,degB1,X,degX,Y,degY,X1,degX1,Y1,degY1] ...

= arya(N,degN,D,degD,gap)

if nargin==4

gap = 1.0e8;

28

end

[B,degB,A,degA] = left_prm(N,degN,D,degD,1,gap);

[A,degA] = transp(A,degA);

[B,degB] = transp(B,degB);

[B1,degB1,A1,degA1,Y,degY,X,degX] = left_prm(B,degB,A,degA,2,gap);

[B1,degB1] = transp(B1,degB1);

[A1,degA1] = transp(A1,degA1);

[X,degX] = transp(X,degX);

[Y,degY] = transp(Y,degY);

[B,degB,A,degA,Y1,degY1,X1,degX1] = left_prm(B1,degB1,A1,degA1,2,gap);

A.2 cindep.m

A.2.1 Purpose

Checks if the last row of a matrix is dependent on the previous rows and if so, returns

the coefficients of dependence. If independent, the coefficients will be null. See Sec. 7.1.

A.2.2 mfiles called

Local routines: makezero.

Routines of mathworks: norm, svd, /.


% function b = cindep( S,gap)

% used in XD + YN = C. all rows except the last of are assumed to

% be independent. The aim is to check if the last row is dependent on the

% rest and if so how. The coefficients of dependence is sent in b

function b = cindep( S,gap)

if nargin == 1

gap = 1.0e8;

end

[rows,cols] = size(S);

if rows > cols

ind = 0;

else

sigma = svd(S);

29

len = length(sigma);

if (sigma(len)/sigma(1) <= (eps*max(i,cols)))

ind = 0; % not independent

else

if any(sigma(1:len-1) ./sigma(2:len)>=gap)

ind = 0; % not dependent

else

ind = 1; % independent

end

end

end

if ind

b = [];

else

b = S(rows,:)/S(1:rows-1,:);

b = makezero(b,gap);

end

A.3 clcoef.m

A.3.1 Purpose

Used to clear trailing zeros.

A.3.2 mfiles called

Kwakernaak’s routines: polsize.

Routines of mathworks: norm, all.


% function [P,degP] = clcoef(Q,degQ)

% MATLAB FUNCTION clcoef TO CLEAR ZERO LEADING

% COEFFICIENT MATRICES OF A POLYNOMIAL MATRIX

% H. Kwakernaak, July, 1990

% Modified by Kannan Moudgalya in Nov. 1992

function [P,degP] = clcoef(Q,degQ)

30

[rQ,cQ] = polsize(Q,degQ);

if all(all(Q==0))

P = zeros(rQ,cQ);

degP = 0;

else

P = Q; degP = degQ; rP = rQ; cP = cQ;

j = degP+1;

while j >= 0

if norm(P(:,(j-1)*cP+1:j*cP),inf) < (1e-8)*norm(P,inf)

P = P(:,1:(j-1)*cP);

degP = degP-1;

else

j = 0;

end

j = j-1;

end

end

A.4 flat2nor.m

A.4.1 Purpose

Rearranges a flat (row) polynomial into a polynomial matrix. Opposite of what nor2flat,

Sec. A.11 does.

A.4.2 mfiles called

Routines of mathworks: error.


% function [X,degX] = flat2nor(flatX,degflatX,rows,bcols)

% returns a flat polynomial X into a polynomial of size rows by bcols

function [X,degX] = flat2nor(flatX,degflatX,rows,bcols)

if length(flatX) == rows * bcols * (degflatX+1)

degX = degflatX;

X = [];

for i = 1:degX+1

31

dummy = zeros(bcols,rows);

dummy(:) = flatX( (i-1)*rows*bcols+1 : i*rows*bcols );

dummy = dummy’;

X = [X dummy];

end

else

error(’flat2nor: dimensions do not agree’)

end

A.5 indep.m

A.5.1 Purpose

Used to check if the rows of a matrix are dependent. If they are dependent, the coefficients

that relate the first dependent row on the previous rows are returned. If the input matrix

is an independent matrix, the coefficient vector will be null. See Sec. 7.1.

A.5.2 mfiles called

Local routines: makezero.

Routines of mathworks: norm, /.


% function b = indep(S,gap)

% determines the first row that is dependent on the previous rows of S.

% The coefficients of dependence is returned in b

function b = indep( S,gap)

if nargin == 1

gap = 1.0e8;

end

[rows,cols] = size(S);

ind = 1;

i = 2;

while ind & i <= rows

sigma = svd(S(1:i,:));

len = length(sigma);

if(sigma(len)/sigma(1) < (eps*max(i,cols)))

ind =0;

32

else

shsig = [sigma(2:len);sigma(len)];

if any( (sigma ./shsig) > gap)

ind = 0;

else

ind = 1;

i = i+1;

end

end

end

if ind

b =[];

else

c = S(i,:)/S(1:i-1,:);

c = makezero(c,gap);

b = [-c 1];

end

A.6 issoln.m

A.6.1 Purpose

Performs the computations indicated in the beginning of Sec. 4.1 and decides whether

there is a solution to Eq. 34.

A.6.2 mfiles called

Local routines: arya.

Kwakernaak’s routines: modified version of l2r.


% function soln = issoln(D,degD,C,degC,B,degB,A,degA)

% finds out if there is a solution to the system XD + YN = C

% [-B A] is the left modified minimal basis of [D’ N’]’

function soln = issoln(D,degD,C,degC,B,degB,A,degA)

[B1,degB1,A1,degA1] = l2r(B,degB,A,degA);

[U1,degU1,R,degR] = l2r(A1,degA1,D,degD);

[C2,degC2,U2,degU2] = left_prm(C,degC,R,degR);

if degU2 == 0

33

soln = 1;

else

soln = 0;

end

A.7 l2r.m

A.7.1 Purpose

Carry out left to right coprime factorization. This is a modified version of Kwakernaak’s

routine.

A.7.2 mfiles called

Local routines: arya.

Kwakernaak’s routines: transp.


% function [Rnum,Rnumdeg,Rden,Rdendeg] = l2r(N,degN,D,degD)

% given Numerator and Denominator polynomial matrices in left form,

% not necessarily coprime, finds right coprime factorisation.

function [Rnum,Rnumdeg,Rden,Rdendeg] = l2r(N,degN,D,degD)

[N,degN] = transp(N,degN);

[D,degD] = transp(D,degD);

[Rnum,Rnumdeg,Rden,Rdendeg] = left_prm(N,degN,D,degD);

[Rnum,Rnumdeg] = transp(Rnum,Rnumdeg);

[Rden,Rdendeg] = transp(Rden,Rdendeg);

A.8 left prm.m

A.8.1 Purpose

Used to produce left coprime factorization given arbitrary right factorization. Used in

all the computations listed in this report.

A.8.2 mfiles called

Local routines: t1calc.

34

Kwakernaak’s routines: rowjoin, polsize, colsplit, clcoef. The last routine has been

modified locally.

Routines of mathworks: nargin, isempty, size, round.


% function [B,degB,A,degA,Y,degY,X,degX] = ...

% left_prm(N,degN,D,degD,job,gap)

%

% does three different things according to integers that ’job’ takes

% job = 1.

% this is the default. It is always done for all jobs.

% -1 -1 -1

% Given ND , returns coprime B and A where ND = A B

% It is enough if one sends the first four input arguments

% If gap is required to be sent, then one can send either 1 or a null

% entry for job

% job = 2.

% first solve for job = 1 and then solve XA + YB = I

% job = 3.

% used in solving XD + YN = C

% after finding coprime factorization, data are returned

%

% convention: the variable with prefix deg stand for degrees

% of the corresponding polynomial matrices

%

% input:

% N: right fraction numerator polynomial matrix

% D: right fraction denominator polynomial matrix

% N and D are not neccessarily coprime

% gap: variable used to zero entries; default value is 1.0e+8

%

% output

% b and A are left coprime num. and den. polynomial matrices

% X and Y are solutions to Aryabhatta identity, only for job = 2

function [B,degB,A,degA,Y,degY,X,degX] = ...

left_prm(N,degN,D,degD,job,gap)

35

if nargin == 4 | nargin == 5

gap = 1.0e8 ;

end

if nargin == 4, job = 1; end

[F,degF] = rowjoin(D,degD,N,degN);

[Frows,Fbcols] = polsize(F,degF); % Fbcols = block columns

Fcols = Fbcols * (degF+1) ; % actual columns of F

T1 = [];pr =[];degT1 = 0; T1rows = 0;shft = 0;

S=F; sel = ones(Frows,1); T1bcols =1;

abar = (Fbcols + 1):Frows; % a_super_bar of B-C.Chang

while isempty(T1) | T1rows < Frows - Fbcols

Srows = Frows*T1bcols; % max actual columns of result

[T1,T1rows,sel,pr] = ...

t1calc(S,Srows,T1,T1rows,sel,pr,Frows,Fbcols,abar,gap);

[T1rows,T1cols] = size(T1);

if T1rows < Frows - Fbcols

T1 = [T1 zeros(T1rows,Frows)];

T1bcols = T1bcols + 1; % max. block columns of result

degT1 = degT1 + 1; %degree of result

shft = shft +Fbcols;

S = seshft(S,F,shft);

sel = [sel;sel(Srows-Frows+1:Srows)];

rowvec = (T1bcols-1)*Frows+(Fbcols+1):T1bcols * Frows;

abar = [abar rowvec]; % A_super_bar of B-C.chang

end

end

[B,degB,A,degA] = colsplit(T1,degT1,Fbcols,Frows-Fbcols);

[B,degB] = clcoef(B,degB);

B = -B;

[A,degA] = clcoef(A,degA);

if job == 2

S = S(sel,:); % columns

[redSrows,Scols] = size(S);

C = [eye(Fbcols) zeros(Fbcols,Scols-Fbcols)]; % append with zeros

T2 = C/S;

T2 = makezero(T2,gap);

T2 = move(T2,find(sel),Srows);

36

[X,degX,Y,degY] = colsplit(T2,degT1,Fbcols,Frows - Fbcols);

[X,degX] = clcoef(X,degX);

[Y,degY] = clcoef(Y,degY);

elseif job == 3

Y = S;

degY = sel;

X = degT1;

degX = Fbcols;

else

if job ~= 1

error(’Message from left_prm:no legal job number specified’)

end

end

A.9 makezero.m

A.9.1 Purpose

Reduces small entries to zeros. See Sec. 7.2.

A.9.2 mfiles called

Routines of mathworks: find, norm.


% function B = makezero(B,gap)

% where B is a vector and gap acts as a tolerance

function B = makezero(B,gap)

if nargin == 1

gap = 1.0e8;

end

temp = B(find(B)); % non zero entries of B

temp = -sort(-abs(temp)); % absolute values sorted in descending order

len = length(temp);

ratio = temp(1:len-1) ./temp(2:len); % each ratio >1

min_ind = min(find(ratio>gap));

our_eps = temp(min_ind+1);

37

zeroind = find(abs(B)<=our_eps);

B(zeroind) = zeros(1,length(zeroind));

A.10 move.m

A.10.1 Purpose

Used in rearranging columns of a matrix.

A.10.2 mfiles called

None.


% function result = move(b,nonred,max)

% Moves matrix b to matrix result with the information on where to move,

% decided by the indices of nonred.

% The matrix result will have as many rows as b has and max number of columns.

% b is augumented with zeros to have nonred number of columns;

% The columns of b put into those of result as decided by nonred.

function result = move(b,nonred,max)

[brows,bcols] = size(b);

b = [b zeros(brows,length(nonred)-bcols)];

result = zeros(brows,max);

result(:,nonred’) = b;

A.11 nor2flat.m

A.11.1 Purpose

Flattens a polynomial matrix into a row vector. Opposite of what flat2nor, Sec. A.4 does.




% function [flatX,degflatX] = nor2flat(X,degX)

% flattens a polynomial X and returns it in flatX. All the polynomials

% are in the notation of Kwakernaak

38

% e.g.

% if X = [x1 (first row)

% x2 (second row)

% ...

% xn] (nth row)

%

% flatX = [x1 x2 ... xn]

function [flatX,degflatX] = nor2flat(X,degX)

[Xrows,Xbcols] = polsize(X,degX);

flatX = [];

for i = 1:degX+1

dummy = X(:,(i-1)*Xbcols+1:i*Xbcols);

dummy = dummy’;

flatX = [flatX dummy(:)’];

end

degflatX = degX;

A.12 polksca.m

A.12.1 Purpose

Performs polynomial ⊗ scalar computation. This is required, for example, in Eq. 49. See

also Sec. A.13.



Routines of mathworks: kron.


% function [kD,degkD] = polksca(D,degD,Mat)

% given polynomial matrix D with degree degD and scalar matrix Mat,

% calculates D kron Mat, where kron stands for kronecker tensor product

function [kD,degkD] = polksca(D,degD,Mat)

[Drows,Dbcols] = polsize(D,degD);

kD = [];

for i = 1:degD+1

39

kD = [kD kron(D(:,(i-1)*Dbcols+1:i*Dbcols),Mat)];

end

degkD = degD;

A.13 scakpol.m

A.13.1 Purpose

Performs scalar ⊗ polynomial computation. This is required, for example, in Eq. 48. See

also polksca.m in Sec. A.12.



Routines of mathworks: kron.


% function [kD,degkD] = scakpol(D,degD,Mat)

% given polynomial matrix D with degree degD and scalar matrix Mat,

% calculates Mat kron D, where kron stands for kronecker tensor product

function [kD,degkD] = scakpol(D,degD,Mat)


kD = [];

for i = 1:degD+1

kD = [kD kron(Mat,D(:,(i-1)*Dbcols+1:i*Dbcols))];

end

degkD = degD;

A.14 seshft.m

A.14.1 Purpose

Position matrix B at the south east corner of A; shift B before positioning. Fill the

undefined entries with zeros.


None.

40


% function C = seshft(A,B,N)

% given A and B matrices, returns C = [<-A-> 0

% 0 <-B->] with B shifted east by N cols

function C = seshft(A,B,N)

[Arows,Acols] = size(A);

[Brows,Bcols] = size(B);

if N >= 0

B = [zeros(Brows,N) B];

Bcols = Bcols + N;

elseif N < 0

A = [zeros(Arows,abs(N)) A];

Acols = Acols +abs(N);

end

if Acols < Bcols

A = [A zeros(Arows,Bcols-Acols)];

elseif Acols > Bcols

B = [B zeros(Brows,Acols-Bcols)];

end

C = [A

B];

A.15 t1calc.m

A.15.1 Purpose

Calculates rows of T1 matrix as discussed in Sec. 2.1.


Local routines: indep, mod

Routines of mathworks: norm, isempty, any,


% function [T1,T1rows,sel,pr] = ...

% t1calc(S,Srows,T1,T1rows,sel,pr,Frows,Fbcols,abar,gap)

% calculates the coefficient matrix T1

41

% redundant row information is kept in sel: redundant rows are marked

% with zeros. The undeleted rows are marked with ones.

function [T1,T1rows,sel,pr] = ...

t1calc(S,Srows,T1,T1rows,sel,pr,Frows,Fbcols,abar,gap)

b = 1; % vector of primary red.rows

while (T1rows < Frows - Fbcols) & any(sel==1) & ~isempty(b)

b = indep( S(sel,:),gap); % send selected rows of S

if ~isempty(b)

b = move(b,find(sel),Srows);

j = length(b);

while ~(b(j) & any(abar==j)) % pick largest nonzero entry

j = j-1; % of coeff. belonging to abar

if ~j

fprintf(’\nMessage from t1calc, called from left_prm\n\n’)

error(’Denominator is noninvertible’)

end

end

if ~any(j<pr & rem(pr,Frows) == rem(j,Frows)) % pr(2),pr(1)

T1 = [T1; b]; % condition is not violated

T1rows = T1rows +1; % accept this vector

end % else don’t accept

pr = [pr; j]; % update prime red row info

while j <= Srows

sel(j) = 0;

j = j + Frows;

end

end

end

A.16 xdnyc.m

A.16.1 Purpose

Solves Eq. 40


Local routines: nor2flat, scakpol, polksca, flat2nor, arya.

42

Kwakernaak’s routines: polsize, transp.


% function [Y,degY,X,degX] = xdnyc(N,degN,D,degD,C,degC)

% solves XD + NY = C

function [Y,degY,X,degX] = xdnyc(N,degN,D,degD,C,degC)

[Crows,Cbcols] = polsize(C,degC);


[N,degN] = transp(N,degN);

[Nrows,Nbcols] = polsize(N,degN);

[Ctilda,degC] = nor2flat(C,degC);

[Dtilda,degD] = scakpol(D,degD,eye(Crows));

[Ntilda,degN] = polksca(N,degN,eye(Drows));

[B,degB,A,degA,Ytilda,degY,Xtilda,degX] = ...

xdync(Ntilda,degN,Dtilda,degD,Ctilda,degC);

[X,degX] = flat2nor(Xtilda,degX,Crows,Drows);

[Y,degY] = flat2nor(Ytilda,degY,Nbcols,Cbcols);

A.17 xdync.m

A.17.1 Purpose

Solves Eq. 34


Local routines: t1calc.

Kwakernaak’s routines: colsplit, clcoef.

Routines of mathworks: norm, size, /.


% function [B,degB,A,degA,Y,degY,X,degX] = xdync(N,degN,D,degD,C,degC,gap)

% given coefficient matrix in T1, primary redundant row information sel,

% solves XD + YN = C

%

function [B,degB,A,degA,Y,degY,X,degX] = xdync(N,degN,D,degD,C,degC,gap)

if nargin == 6

43

gap = 1.0e+8;

end

[F,degF] = rowjoin(D,degD,N,degN);

[Frows,Fbcols] = polsize(F,degF); %Fbcols = block columns

[B,degB,A,degA,S,sel,degT1,Fbcols] = left_prm(N,degN,D,degD,3,gap);

if issoln(D,degD,C,degC,B,degB,A,degA)

[Crows,Ccols] = size(C);

[Srows,Scols] = size(S);

S = S(sel,:);

T2 =[];

for i = 1:Crows,

Saug = seshft(S,C(i,:),0);

b = cindep(Saug);

b = move(b,find(sel),Srows);

T2 =[T2; b];

end

else

error(’message from xdync:No Solution.’)

end

[X,degX,Y,degY] = colsplit(T2,degT1,Fbcols,Frows-Fbcols);

[X,degX] = clcoef(X,degX);

[Y,degY] = clcoef(Y,degY);

References

[1] Astrom, K. J. and Wittenmark, B., “Computer Controlled Systems - Theory and

Design”, Prentice Hall, Englewood Cliffs, NJ 2rd Edition, 1990.

[2] Chang, B-C and Pearson, J. B., “Algorithms for the Solution of Polynomial Equa-

tions arising in Multivariable Control Theory”, Tech. Report No. 8208, Electrical

Engineering Dept., Rice University, May 1982.

[3] Chang, B-C., “Programs for Solving Polynomial Equations” Tech. Report No. 8209,

Electrical Engineering Dept., Rice University, May 1982.

[4] Cheng, L. and Pearson, J. B., “Synthesis of Linear Multivariable Regulators”, IEEE

Trans. Auto. Control, AC-26, pp. 194-202, 1981.

[5] Grimble, M. J. and Kucera, V. (Eds), “Polynomial Methods for Control System

Design”, Springer, London, 1996.

44

[6] Hunt, K. J., “Polynomial Methods in Optimal Control and Filtering”, Peter Pere-

grinus Ltd., London, 1993.

[7] Kailath, T., “Linear Systems”, Prentice-Hall, Inc., Englewood Cliffs, 1980.

[8] Kucera, V., “Discrete Linear Control: The Polynomial Equation Approach”, John

Wiley & Sons, Prague, 1979.

[9] Kwakernaak, H., “MATLAB Macros for Polynomial H∞ Control System Optimiza-

tion”, Memorandum No. 881, Faculty of Applied Mathematics, University of Twente,

Sept. 1990.

[10] Pearson, J. B., “Notes on `1-Optimal Control”, in H∞-Control Theory, Lecture

Notes in Mathematics, 1496, Ed. E. Mosca and L. Pandolfi, Springer-Verlag, Berlin,

1991.

[11] Vidyasagar, M., “Control System Synthesis”, MIT Press, Cambridge, 1985.

[12] Willems, J. C., “Paradigms and Puzzles in the Theory of Dynamical Systems”, IEEE

Trans. Auto. Cont., AC-36, pp. 259-294.

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